The Relativistic Rocket

The theory of relativity sets a severe limit to our ability to explore the galaxy in
space ships. As an object approaches the speed of light, more and more energy is
needed to accelerate it further. To reach the speed of light an infinite amount of
energy would be required. It seems that the speed of light is an absolute barrier
which cannot be reached or surpassed by massive objects (see relativity FAQ article on faster than light travel). Given that the galaxy
is about 100,000 light years across there seems little hope for us to get very far in
galactic terms unless we can overcome our own mortality.

Science fiction writers can make use of worm holes or warp drives to overcome this
restriction, but it is not clear that such things can ever be made to work in
reality. Another way to get around the problem may be to use the relativistic
effects of time dilation and length contraction to cover large distances within a
reasonable time span for those aboard a space ship. If a rocket accelerates at
1g (9.81 m/s2) the crew will experience the equivalent of a
gravitational field with the same strength as that on Earth. If this could be
maintained for long enough they would eventually receive the benefits of the relativistic
effects which improve the effective rate of travel.

What then, are the appropriate equations for the relativistic rocket?

First of all we need to be clear what we mean by continuous acceleration at
1g. The acceleration of the rocket must be measured at any given instant in
a non-accelerating frame of reference travelling at the same instantaneous speed as the
rocket (see relativity FAQ on accelerating clocks). This
acceleration will be denoted by a. The proper time as measured by the crew
of the rocket (i.e. how much they age) will be denoted by T, and the time as
measured in the non-accelerating frame of reference in which they started (e.g. Earth)
will be denoted by t. We assume that the stars are essentially at rest in
this frame. The distance covered as measured in this frame of reference will be
denoted by d and the final speed v. The time dilation or length
contraction factor at any instant is the gamma factor γ.

The relativistic equations for a rocket with constant positive acceleration a >
0 are the following. First, define the hyperbolic trigonometric functions sh, ch,
and th (also known as sinh, cosh, and tanh):

These equations are valid in any consistent system of units such as seconds for time,
metres for distance, metres per second for speeds and metres per second squared for
accelerations. In these units c = 3 × 108 m/s (approx). To
do some example calculations it is easier to use units of years for time and light years
for distance. Then c = 1 lyr/yr and g = 1.03
lyr/yr2. Here are some typical answers for a = 1g.

So in theory you can travel across the galaxy in just 12 years of your own time.
If you want to arrive at your destination and stop then you will have to turn your rocket
around half way and decelerate at 1g. In that case it will take nearly
twice as long in terms of proper time T for the longer journeys; the Earth time
t will be only a little longer, since in both cases the rocket is spending most
of its time at a speed near that of light. (We can still use the above equations to
work this out, since although the acceleration is now negative, we can "run the film
backwards" to reason that they still must apply.)

Here are some of the times you will age when journeying to a few well known space
marks, arriving at low speed:

For distances bigger than about a thousand million light years, the formulas given here
are inadequate because the universe is expanding. General Relativity would have to
be used to work out those cases.

If you wish to pass by a distant star and return to Earth, but you don't need to stop
there, then a looping route is better than a straight-out-and-back route. A good
course is to head out at constant acceleration in a direction at about 45 degrees to your
destination. At the appropriate point you start a long arc such that the centrifugal
acceleration you experience is also equivalent to earth gravity. After 3/4 of a
circle you decelerate in a straight line until you arrive home.

Below the rocket, something strange is happening...

In the rocket, you can make measurements of the world around you. One thing you might
do is ask how the distance to an interesting star you are headed towards changes with
T, the time on your clock. At blast-off (t=T=0) the rocket is at
rest, so this distance initially equals the distance D to the star in the
non-accelerating frame. But once you are moving, however you choose to measure this
distance, it will be reduced by your current distance d travelled in the
non-accelerating frame, as well as the whole lot contracted by a factor of γ, your
Lorentz factor at time T. Eventually you will pass the star and it will
recede behind you. The distance you measure to it at time T is

(D - d)/γ = (D + c2/a)/ch(aT/c) - c2/a

A plot of this distance as a function of T shows that, as expected, it starts
at D, then reduces to zero as you pass the star. Then it becomes negative
as the star moves behind you. As T goes to infinity, the distance
asymptotes to a value of -c2/a. That means that everything in
the universe is falling "below" the rocket, but never receding any farther than a distance
of -c2/a as measured by you. It all piles up just short of this
distance, asymptoting to a plane called a horizon. You see this horizon
actually form as the rocket accelerates, because there comes a time when no signal emitted
from "below" the horizon can ever reach you. Everything falls toward that plane, and
as it does so it begins to redden, due to the increasing red shift of its light, because
you are accelerating. Finally it fades out of visibility. In fact, as anything
gets closer to the horizon, it ages more and more slowly; time comes to a complete halt
there. The horizon is a dark plane that appears to be swallowing everything in the
universe! But of course, nothing strange is noticed by the non-accelerating Earth
observers. There is no horizon anywhere for them.

And inside the rocket, something strange is also happening...

Whereas time slows to a stop a certain distance below the rocket, it speeds up "above"
the rocket (that is, in the direction in which it's travelling). This effect could,
in principle, be measured inside the rocket too: a clock attached to the rocket's ceiling
(i.e. furthest from the motor) ages faster than a clock attached to its floor.

For a standard-sized rocket with a survivable acceleration, this difference in how fast
things age within its cabin is very small. Even so, it tells us something
fundamental about gravity, via Einstein's Equivalence Principle. Einstein
postulated that any experiment done in a real gravitational field, provided that
experiment has a fairly small spatial extent and doesn't take very long, will give a
result indistinguishable from the same experiment done in an accelerating rocket. So
the idea that the rocket's ceiling ages faster than its floor (and that includes the
ageing of any bugs sitting on these) transfers to gravity: the ceiling of the room in
which you now sit is ageing faster than its floor; and your head is ageing faster than
your feet. Earth's rotation complicates this effect, but doesn't alter it
completely.

This difference in ageings on Earth has been verified experimentally. In fact, it
was absolutely necessary to take into account when the GPS satellite system was assembled.

How much fuel is needed?

Sadly there are a few technical difficulties you will have to overcome before you can
head off into space. One is to create your propulsion system and generate the
fuel. The most efficient theoretical way to propel the rocket is to use a "photon
drive". It would convert mass to photons or other massless particles which shoot out
the back. Perhaps this may even be technically feasible if we ever produce an
antimatter-driven "graser" (gamma ray laser).

Remember that energy is equivalent to mass, so provided mass can be converted to 100%
radiation by means of matter-antimatter annihilation, we just want to find the mass
M of the fuel required to accelerate the payload m. The answer is
most easily worked out by conservation of energy and momentum.

First: conservation of energy

The total energy before blast-off is (in the Earth frame)

Einitial = (M+m)c2

At the end of the trip the fuel has all been converted to light with energy
EL, so the total energy is now

Efinal = γmc2 + EL

By conservation of energy these must be equal, so here is our first conservation equation:

(M+m)c2 = γmc2 + EL ........ (1)

Second: conservation of momentum

The total momentum before blast-off is zero in the Earth frame.

pinitial = 0

At the trip's end the fuel has all been converted to light with momentum of magnitude
EL/c, but in the opposite direction to the rocket. So the
final momentum is

pfinal = γ mv - EL/c

By conservation of momentum these must be equal, so our second conservation equation is:

0 = γ mv - EL/c ........ (2)

Eliminating EL from equations (1) and (2) gives

(M+m)c2 - γmc2 = γmvc

so that the fuel:payload ratio is

M/m = γ(1 + v/c) - 1

This equation is true irrespective of how the ship accelerates to velocity v,
but if it accelerates at constant rate a then

This is a lot of fuel—and remember, we are using a motor that is 100% efficient!

What if we prefer to stop at the destination? We accelerate to the half way point
at 1g and then immediately switch the direction of our rocket so that we now
decelerate at 1g for the rest of second half of the trip. The calculations
here are just a little more involved since the trip is now in two distinct halves (and the
equations at the top assume a positive acceleration only). Even so, the answer turns
out to have exactly the same form: M/m = exp(aT/c) - 1, except that the proper time T is
now almost twice as large as for the non-stop case, since the slowing-down rocket is
losing the ageing benefits of relativistic speed. This dramatically increases the
amount of fuel needed:

Compare these numbers to the previous case: they are hugely different! Why should
that be? Let's take the case of Laurel and Hardy, two astronauts travelling to
Vega. Laurel speeds past without stopping, and so only needs 57 kg of fuel for every
1 kg of payload. Hardy wishes to stop at Vega, and so needs 886 kg of fuel for every
1 kg of payload. Laurel takes almost 28 Earth years for the trip, while Hardy takes
29 Earth years. (They both take roughly the same amount of Earth time because they
are both travelling close to speed c for most of the journey.) They travel
neck-and-neck until they've both gone half way to Vega, at which point Hardy begins to
decelerate.

It's useful to think of the problem in terms of relativistic mass, since this is what
each rocket motor "feels" as it strives to maintain a 1g acceleration or
deceleration. The relativistic mass of each traveller's rocket is continually
decreasing throughout their trip (since it's being converted to exhaust energy). It
turns out that at the half way point, Laurel's total relativistic mass (for fuel plus
payload) is about 28m, and from here until the trip's end, this relativistic mass
only decreases by a tiny amount, so that Laurel's rocket needs to do very little
work. So at the halfway point his fuel:payload ratio turns out to be about 1.

For Hardy, things are different. He needs to decrease his relativistic mass to
m at the end where he is to stop. If his rocket's total relativistic mass
at the halfway point were the same as Laurel's (28m), with a fuel:payload ratio
of 1, Hardy would need to decrease the relativistic mass all the way down to m at
the end, which would require more fuel than Laurel had needed. But Hardy wouldn't
have this much fuel on board—unless he ensures that he takes it with him
initially. This extra fuel that he must carry from the start becomes more payload (a
lot more), which needs yet more fuel again to carry that. So suddenly his fuel
requirement has increased enormously. It turns out that at the half way point, all
this extra fuel gives Hardy's rocket a total relativistic mass of about 442m, and
his fuel:payload ratio turns out to be about 29.

Another way of looking at this odd situation is that both travellers know that they
must take fuel on board initially to push them at 1g for the total trip time.
They don't care about what's happening outside. In that case, Laurel travels for 28
Earth years but ages just 3.9 years, while Hardy travels for 29 Earth years but ages 6.6
years. So Hardy has had to sit at his controls and burn his rocket for almost twice
as long as Laurel, and that has required more fuel, with even more fuel required because
of the fuel-becomes-payload situation that we mentioned above.

This fuel-becomes-payload problem is well known in the space programme: part of the
reason the Saturn V moon rocket was so big was because it needed yet more fuel just to
carry the fuel it was already carrying.

Other fuel options

Well, this is probably all just too much fuel to contemplate. There are a limited
number of solutions that don't violate energy-momentum conservation or require
hypothetical entities such as tachyons or worm holes.

It may be possible to scoop up hydrogen as the rocket goes through space, using fusion
to drive the rocket. This would have big benefits because the fuel would not have to
be carried along from the start. Another possibility would be to push the rocket
away using an Earth-bound grazer directed onto the back of the rocket. There are a
few extra technical difficulties but expect NASA to start looking at the possibilities
soon :-).

You might also consider using a large rotating black hole as a gravitational catapult
but it would have to be very big to avoid the rocket being torn apart by tidal
forces or spun at high angular velocity. If there is a black hole at the centre of
the Milky Way, as some astronomers think, then perhaps if you can get that far, you can
use this effect to shoot you off to the next galaxy.

One major problem you would have to solve is the need for shielding. As you
approach the speed of light you will be heading into an increasingly energetic and intense
bombardment of cosmic rays and other particles. After only a few years of
1g acceleration even the cosmic background radiation is Doppler shifted into a
lethal heat bath hot enough to melt all known materials.

For the derivation of the rocket equations see "Gravitation" by Misner, Thorne and
Wheeler, Section 6.2.